Normal form for codimension two Levi-at CR singular submanifolds

Normal form for codimension two Levi-at CR singular submanifolds Jir Lebl 1, Xianghong Gong 2 1 Department of Mathematics, Oklahoma State University 2 Department of Mathematics, University of Wisconsin{Madison May 21, 2014 Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 1 / 22

Bishop surfaces M C 2, CR singular submanifold of codimension two with a nondegenerate complex tangent: w = z z + (z 2 + z 2 ) + O(3) if 2 [0; 1) and for = 1 we have w = z 2 + z 2 + O(3): < 1 2 is elliptic, = 1 2 is parabolic, and > 1 2 is hyperbolic. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 2 / 22

If 0 < < 1 2, Moser-Webster found the normal form = 1; 0, and s 2 N. w = z z + + (Re w) s (z 2 + z 2 ) Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 3 / 22

If 0 < < 1 2, Moser-Webster found the normal form = 1; 0, and s 2 N. w = z z + + (Re w) s (z 2 + z 2 ) In the other cases, things (especially convergence) become more dicult, but are now fairly well understood. (Huang, Huang-Krantz, Moser, Gong, Huang-Yin, Coman, Kenig-Webster,... ) Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 3 / 22

If 0 < < 1 2, Moser-Webster found the normal form = 1; 0, and s 2 N. w = z z + + (Re w) s (z 2 + z 2 ) In the other cases, things (especially convergence) become more dicult, but are now fairly well understood. (Huang, Huang-Krantz, Moser, Gong, Huang-Yin, Coman, Kenig-Webster,... ) When = 0, w = z z ; w = z z + z s + z s + O(s + 1); there are innitely many formal biholomorphic invariants (Moser, Huang-Krantz, Huang-Yin). Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 3 / 22

Codimension two Let M C n+1 be a codimension 2 real-analytic submanifold with a nondegenerate complex tangent. Let M CR be the set of CR points. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 4 / 22

Codimension two Let M C n+1 be a codimension 2 real-analytic submanifold with a nondegenerate complex tangent. Let M CR be the set of CR points. In C 2, M CR has \no CR structure"; i.e. M CR is totally real. In C 3, M CR has CR invariants in C 3. Normal form for M will be at least as hard (probably harder) than a local normal form for CR submanifolds. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 4 / 22

Codimension two Let M C n+1 be a codimension 2 real-analytic submanifold with a nondegenerate complex tangent. Let M CR be the set of CR points. In C 2, M CR has \no CR structure"; i.e. M CR is totally real. In C 3, M CR has CR invariants in C 3. Normal form for M will be at least as hard (probably harder) than a local normal form for CR submanifolds. Various cases of codimension two CR singular manifolds have been studied by Huang-Yin, Dolbeault-Tomassini-Zaitsev, Burcea, Coman, Slapar,... Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 4 / 22

No CR invariants A codimension 2 submanifold with \trivial CR structure" in C n+1? Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 5 / 22

No CR invariants A codimension 2 submanifold with \trivial CR structure" in C n+1? How about Levi-at! Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 5 / 22

No CR invariants A codimension 2 submanifold with \trivial CR structure" in C n+1? How about Levi-at! A CR submanifold is Levi-at if the Levi-form (Levi-map?) vanishes. It is standard that such a (real-analytic) submanifold is locally Im z 1 = 0; Im z 2 = 0: There are no holomorphic invariants. In C 2 the notions coincide. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 5 / 22

Foliation of Levi-ats Take M to be Im z 1 = 0; Im z 2 = 0: M is foliated by complex submanifolds: x z 1 and z 2 at some real-value. (The Levi-foliation) Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 6 / 22

Foliation of Levi-ats Take M to be Im z 1 = 0; Im z 2 = 0: M is foliated by complex submanifolds: x z 1 and z 2 at some real-value. (The Levi-foliation) The foliation extends (uniquely) to a holomorphic foliation of a neighborhood: leaves are obtained by xing z 1 and z 2. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 6 / 22

Our class of submanifolds Let M C n+1, n 2, be a real-analytic CR singular submanifold with a nondegenerate complex tangent at 0, such that M CR is Levi-at. We want the normal form for such manidolds. Take (z ; w) 2 C n C. Write M as w = (z ; z ) for a real-analytic complex-valued function vanishing to second order at the origin. (It is really two real equations). Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 7 / 22

Detour: mixed-holomorphic submanifolds For a holomorphic f take X C m given by f (z 1 ; : : : ; z m 1 ; z m ) = 0: X is codimension 2. (We can think of it as a complex analytic subvariety, thinking of z m as another holomorphic coordinate, but then our automorphism group is dierent). If X is a submanifold, it is Levi-at at all CR points, and it can clearly be CR singular. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 8 / 22

Detour: mixed-holomorphic submanifolds For a holomorphic f take X C m given by f (z 1 ; : : : ; z m 1 ; z m ) = 0: X is codimension 2. (We can think of it as a complex analytic subvariety, thinking of z m as another holomorphic coordinate, but then our automorphism group is dierent). If X is a submanifold, it is Levi-at at all CR points, and it can clearly be CR singular. Exercise, suppose m = 2: Classify all such submanifolds locally up to local biholomorphisms. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 8 / 22

Quadratic parts In the following let M C n+1, n 2, be a germ of a real-analytic real codimension 2 submanifold, CR singular at the origin, written in coordinates (z ; w) 2 C n C as w = A(z ; z ) + B(z ; z ) + O(3); for quadratic A and B, where A + B 6 0 (nondegenerate complex tangent). Suppose M is Levi-at (that is M CR is Levi-at). Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 9 / 22

Quadratic parts Theorem (i) If M is a quadric, then M is locally biholomorphically equivalent to one and exactly one of the following: (A.1) w = z 1 2 ; (A.2) w = z 1 2 + z 2 2 ;. (A.n) w = z 2 1 + z 2 2 + + z 2 n ; (B.) w = jz 1 j 2 + z 2 1 ; 0; (C.0) w = z 1 z 2 ; (C.1) w = z 1 z 2 + z 1 2 : Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 10 / 22

Quadratic parts Theorem (ii) For general M the quadric w = A(z ; z ) + B(z ; z ) + O(3) w = A(z ; z ) + B(z ; z ) is Levi-at, and can be put via a biholomorphic transformation into exactly one of the forms above. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 11 / 22

Bishop-like The quadrics (A.1) w = z 2 1 ; (B.) w = jz 1 j 2 + z 2 1 ; 0: These are of the form N C n 1 for a Bishop surface N C 2. Not every M with quadratic part of type A.1 or B. is of the form N C n 1. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 12 / 22

Bishop-like The quadrics (A.1) w = z 2 1 ; (B.) w = jz 1 j 2 + z 2 1 ; 0: These are of the form N C n 1 for a Bishop surface N C 2. Not every M with quadratic part of type A.1 or B. is of the form N C n 1. If the Levi-foliation of M extends to a non-singular holomorphic foliation of a neighborhood of the origin, then M is either type A.1 or B. and can be written as N C n 1. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 12 / 22

Nondegeneracy We consider C.1 the \nondegenerate case." w = A(z ; z ) + B(z ; z ): The form A \represents the Levi-form." A can have rank at most 2 (actually 1) for M to be Levi-at. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 13 / 22

Nondegeneracy We consider C.1 the \nondegenerate case." w = A(z ; z ) + B(z ; z ): The form A \represents the Levi-form." A can have rank at most 2 (actually 1) for M to be Levi-at. For type A.k, the form A = 0, and so we consider these degenerate. These can be considered as a generalization to higher dimension of Bishop surfaces with Bishop invariant = 1. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 13 / 22

Nondegeneracy We consider C.1 the \nondegenerate case." w = A(z ; z ) + B(z ; z ): The form A \represents the Levi-form." A can have rank at most 2 (actually 1) for M to be Levi-at. For type A.k, the form A = 0, and so we consider these degenerate. These can be considered as a generalization to higher dimension of Bishop surfaces with Bishop invariant = 1. For type B., the form A is real-valued and so we also consider it degenerate. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 13 / 22

Nondegeneracy We consider C.1 the \nondegenerate case." w = A(z ; z ) + B(z ; z ): The form A \represents the Levi-form." A can have rank at most 2 (actually 1) for M to be Levi-at. For type A.k, the form A = 0, and so we consider these degenerate. These can be considered as a generalization to higher dimension of Bishop surfaces with Bishop invariant = 1. For type B., the form A is real-valued and so we also consider it degenerate. Only C.x has a complex-valued A, and C.1 also has a nonzero B. These have no analogue in C 2. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 13 / 22

Stability Only C.1 and A.n are stable under perturbation (preserving Levi-atness, and CR singularity) CR singularities generally not isolated and can change in type from point to point: Example: w = z 2 1 + z 1 z 2 z 3 ; is type A.1 at the origin, but of type C.1 at nearby CR singular points. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 14 / 22

Quadrics in C n+1 Type CR singularity S dim R S S A.k z 1 = 0,..., z k = 0, w = 0 2n 2k complex B. 1 2 z 1 + z 1 = 0, w = 0 2n 1 Levi-at B., 6= 1 2 z 1 = 0, w = 0 2n 2 complex C.0 z 2 = 0, w = 0 2n 2 complex C.1 z 2 + 2z 1 = 0, w = z 2 2 4 2n 2 Levi-at Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 15 / 22

Foliations and C.x Theorem Suppose M C n+1, n 2, is a real-analytic Levi-at CR singular submanifold of type C.1 or C.0, that is, w = z 1 z 2 + z 2 1 + O(3) or w = z 1 z 2 + O(3): Then there exists a nonsingular real-analytic foliation dened on M that extends the Levi-foliation on M CR, and consequently, there exists a CR real-analytic mapping F : U R 2 C n 1! C n+1 such that F is a dieomorphism onto F (U ) = M \ U 0, for some neighborhood U 0 of 0. Really a Nash blowup, see also a related paper by Garrity. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 16 / 22

Note that the Levi-foliation does not always extend (even to M only) for the other types. Example: A.2: w = z 2 1 + z 2 2 The \leaf" of the foliation becomes singular at the origin. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 17 / 22

Mixed-holomorphic C.1 For mixed holomorphic C.1, we completely understand the situation. In this case we can set things up to use implicit function theorem. Theorem Let M C n+1, n 2, be a real-analytic submanifold given by w = z 1 z 2 + z 2 1 + r(z 1 ; z 1 ; z 2 ; z 3 ; : : : ; z n ); where r is O(3). Then M is Levi-at and at the origin M is locally biholomorphically equivalent to the quadric M C :1 submanifold w = z 1 z 2 + z 2 1 : Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 18 / 22

General normal form for C.1 Could things be really this simple in general? Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 19 / 22

General normal form for C.1 Could things be really this simple in general? No. There are in fact innitely many formal invariants. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 19 / 22

General normal form for C.1 Could things be really this simple in general? No. There are in fact innitely many formal invariants. Theorem Let M be a real-analytic Levi-at type C.1 submanifold in C 3. There exists a formal biholomorphic map transforming M into the image of ^'(z ; z ; ) = z + A(z ; ; w)w ; ; w with = z + 1 2 and w = z + z 2. Here A = 0, or A satises certain normalizing conditions. When A 6= 0 the formal automorphism group preserving the normal form is nite or 1 dimensional. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 19 / 22

Automorphisms of the C.1 quadric Suppose M C 3 w = z 1 z 2 + z 2 1 ; and (F 1 ; F 2 ; G) is a local automorphism at the origin, then F 1 depends only on z 1, F 2 and G depend only on z 2 and w, and F 1 completely determines F 2 and G. On the other hand, given any F 1 with F 1 (0) = 0, there exist unique F 2 and G that complete an automorphism. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 20 / 22

Automorphisms of the C.1 quadric Suppose M C 3 w = z 1 z 2 + z 2 1 ; and (F 1 ; F 2 ; G) is a local automorphism at the origin, then F 1 depends only on z 1, F 2 and G depend only on z 2 and w, and F 1 completely determines F 2 and G. On the other hand, given any F 1 with F 1 (0) = 0, there exist unique F 2 and G that complete an automorphism. In higher dimensions the extra components of the mapping are arbitrary. Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 20 / 22

Involution on the C.1 quadric The proofs use the following key fact: For the C.1 quadric w = z 1 z 2 + z 1 2 we have the involution (z 1 ; z 2 ; : : : ; z n ; w) 7! ( z 2 z 1 ; z 2 ; : : : ; z n ; w) Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 21 / 22

Thank you! Jir Lebl, Xianghong Gong Levi-at CR singular submanifolds May 21, 2014 22 / 22